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Pi And Ramanajan

Someone has finally complained about an equality sign in SF#37 namely,

22[PI]4 = 2143

D. Thomas has correctly pointed out that we have here only a very good approximation. Of course, one need not do the actual calculation to prove that it is an approximation, because 2143/22 is a rational fraction which can be expressed as a repeating decimal; whereas pi is irrational.

The number (2143/22)¼ is a discovery of Ramanujan, about whom we heard on p. 000. How did he ever stumble upon this extremely accurate approximation of pi -- one that is accurate to 300 parts in a trillion? N.D. Mermin suggests that Ramanujan may have taken it from the expansion:

[PI]4 = 97 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(16539 +....

If 16,539 is replaced by infinity, Ramanujan's result follows.

(Mermin, N. David; "Pi in the Sky," American Journal of Physics, 55:584, 1987.)